\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 177, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/177\hfil Energy quantization]
{Energy quantization for approximate H-surfaces and applications}
\author[S. Zheng \hfil EJDE-2013/177\hfilneg]
{Shenzhou Zheng}
\address{Shenzhou Zheng \newline
Department of Mathematics, Beijing Jiaotong University,
Beijing 100044, China}
\email{shzhzheng@bjtu.edu.cn, Phone +86-10-51688449}
\thanks{Submitted February 2, 2013. Published July 30, 2013.}
\subjclass[2000]{35J50, 35K40, 58D15}
\keywords{Approximate H-surface maps; energy quantization;
H-surface flows; \hfill\break\indent concentration of energy;
bubbling phenomena}
\begin{abstract}
We consider weakly convergent sequences of approximate H-surface maps
defined in the plane with their tension fields bounded in $L^p$ for
$p> 4/3$, and establish an energy quantization that accounts for
the loss of their energies by the sum of energies over finitely many
nontrivial bubbles maps on $\mathbb{R}^2$. As a direct consequence,
we establish the energy identity at finite singular time to their
H-surface flows.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}
The main aim of this study is to discuss the energy quantization
of weakly convergent sequences for the weak solutions of approximate
H-surface maps. Similar to approximate harmonic or biharmonic maps
with the controlled tension or bi-tension fields
\cite{DT,QT,LW1,WZ1,WZ2}, we consider energy quantization of approximate
H-surface maps not only its own interest but also an important application
to H-surface flows. In fact, As a direct consequence we will show energy
identity to so called H-surface flows.
Let $\Omega\subset \mathbb{R}^2$ be a bounded smooth domain, and
$H:\mathbb{R}^3\to \mathbb R$ be a given bounded measurable
function; i.e., $H(\cdot)\in L^{\infty}(\mathbb{R}^3) $.
First we recall the notion of approximate H-surface maps.
\begin{definition}\rm
A map $u\in W^{1,2}(\Omega,\mathbb R^3)$ is called an approximate
H-surfaces, if there exists a tension
field $\tau\in L^p_{\rm{loc}}(\Omega, \mathbb R^{3}), p\ge 1$ such that
\begin{equation}\label{approx_H}
\tau(u)=\Delta u - 2H(u)u_x\wedge u_y,\quad\text{in }\Omega.
\end{equation}
In particular, if $\tau\equiv0$, then the map $u$ satisfies
\begin{equation}\label{H_surfaces}
\Delta u = 2H(u)u_x\wedge u_y ,\quad\text{in }\Omega
\end{equation}
which is called a $H$-surface.
\end{definition}
It is well-known that if $u$ is a conformal representation of a surface
$\mathcal{S}=u(\Omega)$; i.~e.,
\[ %\label{conformal_repres}
\|u_x\|^2- \|u_y\|^2=u_x \cdot u_y=0 ,
\]
then $H(u)$ is the mean curvature of the surface $\mathcal{S}$ at the point $u$.
Notice that H-surface is a critical point of the following energy
functional in $W^{1,2}(\Omega, \mathbb R^3)$.
\begin{equation}\label{H functional}
J_H(u) := \int_{\Omega} \Big(|\nabla u|^2 +\frac 43 Q(u)u_x\wedge u_y\Big)
\end{equation}
with
$$
Q(u)= \Big(\int_{0}^{u_1} H(s, u_2,u_3)ds,\int_{0}^{u_2} H(u_1, s, u_3)ds,
\int_{0}^{u_3}H(u_1, u_2, s)ds\Big).
$$
From the view of geometrical significance, the system \eqref{H_surfaces}
can be regarded to be the minimization problem of the standard energy
$E(u,\Omega):=\int_{\Omega} |\nabla v|^2$ with a constraint of the
prescribed volume $V(v):=\frac 13 \int_{\Omega} v\cdot v_x\wedge v_y
=\text{Constant}$; that is,
\begin{equation}
\min_{v\in W^{1,2}(\Omega,R^3)}\Big\{\int_{\Omega} |\nabla v|^2: v
=\phi \text{ on }\partial\Omega, V(v)=C\Big\},
\end{equation}
for any given $\phi\in W^{1,2}(\Omega)$, and here $H$ is so-called
Lagrangian multiplier.
Wente \cite{Wente} and Hildebrandt \cite{Hil} made fundamental contributions
on the existence of solutions to the planar Plateau problem or surfaces
with constant mean curvature, respectively
(see also Helein's monograph \cite{Hel}). Later, Brezis-Coron \cite{BC1}
and Struwe \cite{St} showed existence of multiple solutions of H-surface
maps in a bounded domain of $\mathbb{R}^2$ for given boundary data.
As we knew, for variable $H$ there were many significant works by
Rey \cite{Re}, Bethuel-Rey \cite{BR}, Caldiroli-Musina \cite{CM} and
Chen-Levine \cite{CL}. Meanwhile, the regularity and bubbling phenomena
analysis to so-called H-surface flows in $ W^{1,2}(\Omega, \mathbb{R}^3)$
has been shown in various cases such as $H$ is a constant, H depends only
on two variables, or $H(u)$ is uniformly Lipschizt continuous
(see Brezis-Coron\cite {BC} and Hong-Hsu \cite{HH}). In addition, for the
high dimensional case ($n>2$), Mou-Yang \cite{MY} introduced H-systems in
a bounded domain of $\mathbb R^n$ and established the existence of multiple
solutions of H-system for a constant $H$ and given boundary data.
Furthermore, Duzaar-Grotowski \cite{DG} studied the existence of solutions
of the H-system with a variable function H from a domain into a higher
dimensional compact Riemannian manifold. All in all, it is an important
observation that H-surface maps are invariant under the dilation
transformations in $\mathbb R^2$. Such a property leads to non-compactness
of sequences of H-surfaces in $\mathbb R^2$, which prompts studies
by Brezis-Coron \cite{BC1} concerning the failure of strong convergence
for weakly convergent H-surfaces. Roughly speaking, the results in \cite{BC1}
assert that the failure of strong convergence occurs at finitely many
concentration points of its energy, where finitely many bubbles
(i. e. any nontrivial solutions in $\mathbb R^2$) are generated, and the
total energies from these bubbles account for the total loss of its energies
during the process of convergence.
Based on the above observation, our main purpose is to extend the results
from \cite{BC1,Re,HH} to the context of suitable approximate H-surface
maps due to its more flexible applications. More precisely, we have
\begin{theorem}\label{energy_identity}
Let $\Omega\subset\mathbb{R}^2$ be a bounded smooth domain. Suppose
that $\{u^k\}_{k=1}^{\infty}\subset W^{1,2}(\Omega,\mathbb{R}^3)\cap
L^{\infty}(\Omega,\mathbb{R}^3)$ is a sequence of approximate
H-surface satisfying
\begin{equation}\label{approx_sequence}
\Delta u^k = 2H(u^k)u^k_x\wedge u^k_y +f^k
\end{equation}
with $H(\cdot)\in L^{\infty}(\mathbb{R}^3)$ and $f^k(x)\in
L^p(\Omega)$ with $p>4/3 $. Let
\begin{equation}\label{uniform-bound}
\sup_{k\in \mathbb{N}}\Big(\|\nabla
u^k\|_{L^2(\Omega)}+ \|H(u^k)\|_{L^{\infty}(\mathbb{R}^3)}
+\|f^k\|_{L^p(\Omega)}\Big)\le M 4/3$ through so called Pohozaev argument \cite{LW1} \cite{WZ1}.
During using control the radial energy by the angular hessian energy and
$L^p$-norm of its tension fields by Pohozaev argument in the neck region,
the assumption $p>4/3$ seems to be necessary to validate the Pohozaev
argument, since we need
$\Delta u^k\cdot(x\cdot\nabla u^k) \in L^1$ and
$f^k\cdot(x\cdot\nabla u^k)\in L^1$.
A typical application of Theorem \ref{energy_identity} is to study asymptotic
behavior at finite time for H-surface flows in the plane.
We can directly obtain identity energy at finite time to H-surface
flows with initial data $u_0$ as follows.
\begin{equation}\label{flow-pro}
\begin{gathered}
u_t=\Delta u - 2H(u)u_x\wedge u_y, \quad (x,t)\in \Omega\times (0,+\infty)\\
u|_{t=0}=u_0,\quad \quad x\in \Omega\\
u|_{\partial\Omega}=u_0|_{\partial\Omega}, \quad t>0, x\in \partial\Omega
\end{gathered}
\end{equation}
where $u_0\in W^{1,2}(\Omega) $ and $H\in L^{\infty}(\mathbb{R}^3)$.
In particular, note that any $t$-independent solution
$u:\Omega \to \mathbb{R}^3$ of \eqref{flow-pro} is a H-surface system.
We are inspired by Hong-Hsu's energy inequality in \cite[Theorem 3.7]{HH}:
for arbitrary $u_0\in C^2(\Omega,\mathbb{R}^3)$ satisfying
$\|u_0\|_{L^{\infty}}\|H\|_{L^{\infty}}<1$,
there exists a time $T_0 > 0$ such that
\begin{equation}\label{energy_ineq}
\|u_t\|_{L^2(\Omega\times(0,T_0))}\le J_H[u_0],
\end{equation}
where $J_H$ is represented by \eqref{H functional}.
Then we also consider that, for a finite singular time $T_0 < +\infty$,
energy identity accounting for the $ \delta$ mass by finite many bubbles.
This observation can be proved by applying the rescaled maps to conformal
invariance of H-surface flows. Then, from the energy inequality
\eqref{energy_ineq} there exists a sequence $t_k\uparrow T_0$ such
that $u^k:=u(\cdot, t_k)\in W^{1,2}(\Omega,\mathbb{R}^3)$ satisfies
\begin{itemize}
\item[(i)] $ \tau_2(u^k):=\|u_t(t_k)\|_{L^2}\to 0$; and
\item[(ii)] $u^k$ satisfies in the distribution sense
\begin{equation}\label{approx_H-surf-1}
\Delta u^k = 2H(u^k)u^k_x\wedge u^k_y +\tau_2(u^k).
\end{equation}
\end{itemize}
Therefore, from Theorem \ref{energy_identity} we derive that an energy
identity of the weak limit of H-surface flows are connected together
without any neck region. In particular, the image of $u_n$
converges pointwise to the image of the limit bubble tree maps,
which is similar to harmonic map flows ( \cite{DT} \cite{QT} \cite{LW1}).
More precisely, we have the following theorem.
\begin{theorem}\label{flow_identity}
For some $T_0 < +\infty$, let
$u\in C^{2+\alpha,1+\alpha}_{\rm loc}(\Omega\times (0, T_0))$
be a solution to \eqref{flow-pro} with
$\|u_0\|_{L^{\infty}(\Omega)}\|H\|_{L^{\infty}(\Omega)}<1$, where
$T_0$ is a singular time. Then there exist a finite many bubbles
$\omega_i,i=1,\dots,L $ such
that
\begin{equation}\label{energy_id2}
\lim_{t\to T_0}E(u(\cdot,t),\Omega)=E(u(\cdot,T_0),\Omega)
+\sum_{j=1}^{L}E(\omega_i,\mathbb S^2),
\end{equation}
where $E(\cdot,\mathbb S^2)$ is the energy of finite many bubbles on
the unit sphere $\mathbb S^2$.
\end{theorem}
The article is organized as follows. In \S 2, we establish a
locally H\"oler continuity of weak solutions and the higher
integrability of their first and second order derivatives,
strong convergence and blow-up analysis to any approximate H-surface
maps with the smallness energy condition and its tension field in $L^p$
for some $p>1$. In \S 3, we prove main Theorem \ref{energy_identity}
by establishing that there is no concentration of angular energy in
the neck region; and then controlling the radial energy in the neck region
by angular energy and $L^p$-norm of its tension field with $p>4/3$
by the Pohozaev argument.
In \S 4, As a consequence of the main Theorem, we set up the
energy identity at finite singular time $T_0 < +\infty$ to sequences of
H-surface flows.
\section{A priori estimates of approximate H-surfaces}
This section is mainly devoted to a locally H\"older continuity of
weak solutions and the higher integrality of the first and second-order
derivatives under the smallness energy. To this end, we need to use
Riesz potential estimates in the Morrey spaces due to Adams \cite{A}.
For an open set $U\subset\mathbb R^n$, $1\le p0$,
which satisfies
$$
\|f\|^p_{L^{p,*}(\Omega)}:= \sup_{t>0}t^p|\{x\in \Omega|
|f(x)|>t\}|0, 0 0}\Big\{x\in \Omega: \lim\inf_{k}\int_{B_r(x)}|\nabla
u^k|^2dy\ge \varepsilon^2_0 \Big\}.
\end{equation}
Let $\Sigma_1 = \{x_1,x_2,\dots,\}$ be any discrete points of
$\Sigma$, and $\{B_{\delta_0} (x_i)\}_{i=1}^\infty $ be mutually
disjoint balls for small $\delta_0$. Then we have
$$
\lim\inf_{k}\int_{B_r(x_i)}|\nabla u^k|^2dy\ge
\varepsilon^2_0,\quad \forall\ 1\le i\le \infty.
$$
Therefore, there exists a natural number $K$ such that for $k\ge K$
we have
$$
\int_{B_r(x_i)}|\nabla u^k|^2dy\ge \varepsilon^2_0,\quad \forall\
1\le i\le \infty.
$$
Let $\mathcal{H}_0$ denote the 0-dimensional Hausdorff measure, then
\begin{equation}\label{finite-sigma}
\begin{aligned}
\varepsilon^2_0 \mathcal{H}_0(\Sigma)
&\leq \sum_{i=0}^{\infty}\int_{B_r(x_i)}|\nabla u^k|^2dy\\
&= H_0(\Sigma)\int_{\cup_{i=0}^{\infty} B_r(x_i)}|\nabla u^k|^2dy\\
&\leq \int_{\Omega}|\nabla u^k|^2dy\le N2) $ due
to Lemma \ref{integ_smallness}, therefore, for any compact subset
$K\subset \Omega\setminus \Sigma$ it
follows from a simple covering argument that $\nu(K) = 0$ and
$u^k\to u$ strongly in $W^{1,2}(K,\mathbb{R}^3)$. Moreover, for any $x_0\in
K$ there is a $r_{0}> 0$ such that
$$
\lim_k \int_{B_{r_0} (x_0)} |\nabla u^k|^2\le \varepsilon^2_0.
$$
By the standard diagonal process we can extract a subsequence of
$u^k$, still denoted as itself, such that $u^k\to u$ in
$W^{1,2}(\Omega\setminus \{x_1,\dots,x_L\}, \mathbb{R}^3)\cap C^{0}(\Omega\setminus
\{x_1,\dots,x_L\}), \mathbb{R}^3)$. Hence, it is easy to see that
the expression \eqref{energy_id1} holds with ``$=$'' replaced by ``$\ge$''.
To prove ``$\le$'' of \eqref{energy_id1},
we need to show that the $L^{2}$-norm of $\nabla u^k$ over any
neck region is arbitrarily small. This will mainly be done in the next
sections. Therefore, we will return to the proof of
Theorem \ref{energy_identity} (2) in the next section.
\section{No concentration of energy in the neck region}
In this section, we show that there is no concentration of
$\|\nabla u^k\|_{L^{2}}$ in the neck region. This will be done in two steps:
the first step is to show that there is no angular energy concentration
in the neck region by comparing with radial harmonic functions over
dyadic annulus. The second step is to control the radial component of
energy by the angular component of energy by way of the Pohozaev argument.
\subsection*{Proof of Theorem \ref{energy_identity} (2)}
Without loss of generality, we suppose that
$\{u^k \}\subset W^{1,2}(B_1,\mathbb{R}^3)$ is a sequence of approximate
H-surface maps with
\begin{equation}\label{uniform-bound2}
\sup_{k\in \mathbb{N}}\Big(\|\nabla
u^k\|_{L^2(B_1)}+ \|H(u^k)\|_{L^{\infty}(\mathbb{R}^3)}+\|f^k\|_{L^p(B_1)}\Big)
\le M,
\end{equation}
which satisfy $u^k\rightharpoonup u$ in $W^{1,2}(B_1)$, $f^k\rightharpoonup f$
in $L^p(B_1)$, and $u^k\to u $ in $W^{1,2}_{\rm loc}(B_1\setminus \{0\})$
but not in $W^{1,2}(B_1)$. In according of Ding and Tian \cite{DT}, we may
assume that the total number of bubbles generated at $0$ is $L=1$.
Then, for any $\epsilon>0$, there is $r_k\downarrow 0, R>1$ large enough
and $0\frac 43$. That is due to $\frac {2p}{2-p}>\frac p{p-1} $ at this point.
On the other hand, thanks to Equ.\eqref{approx_sequence} it implies that
$(\Delta u^k-f^k)=H(u^k)u^k_x\wedge u^k_y\perp T_{u^k(x)}\mathcal{S}$, a. e.
$x\in B_{\delta}$ with $\mathcal{S}=u^k(B_{\delta})$. Therefore,
by multiplying \eqref{approx_sequence} by $x\cdot \nabla u^k$ and
integrating it over $B_{r}$, for $00$, we know from \eqref{bubble-number} that
$u^k(\cdot,\eta_k)\rightharpoonup v$ weakly in $W^{1,2}(B_R, \mathbb{R}^3)$.
We claim that $v$ is a constant map. Indeed, let $|t_k|\le 2\lambda^2_k$ we
observe that
$$
\int_{B_{R}}\Big|u^k(\cdot,\eta_k)-u^k(\cdot,-t_k\lambda^{-2}_k) \Big|^2
\le 4 \int^{2}_{-2}\int_{B_{\lambda^{-1}_k}}\Big|\frac{\partial u^k}{\partial t}
\Big|^2\to 0,
$$
and
$$
\int_{B_{R}}\Big|\nabla u^k(\cdot,-t_k\lambda^{-2}_k) \Big|^2
=\int_{B_{\lambda_kR}}\Big|\nabla u\Big|^2(\cdot,T_0)\to 0.
$$
For each $R>0$, now we apply Theorem \ref{energy_identity} of approximate H-surfaces
to $u^k(\cdot,\eta_k)$ on the ball $B_R$ to conclude that there exist
finite number bubbles $\{\omega_{i,R}\}_{i=1}^{L_R}$ such that
\begin{equation}\label{i-sum-bubbles}
\lim_{k\to \infty}\int_{B_{R}}\Big|\nabla u^k\Big|^2(\cdot,\eta_k)
=\sum_{i=1}^{L_R}E(\omega_{i,R},\mathbb{S}^2).
\end{equation}
Further, we know that $1\le L_R\le \Big[\frac {m}{\varepsilon_0}\Big]$
because there is a $\varepsilon_0$ such that any bubble
$\omega:\mathbb{S}^2\to \mathbb{R}^3$ satisfying
$E(\omega,\mathbb{S}^2)\ge \varepsilon_0$. Hence, there exists a
$d\in \Big[1,\frac {m}{\varepsilon_0}\Big]$ such that, after possible
a subsequence, $L_R=d$ and
\begin{equation}\label{m-bubbles}
m=\lim_{R\uparrow\infty}\lim_{k\to \infty}\int_{B_{R}}\Big|
\nabla u^k\Big|^2(\cdot,\eta_k)=\lim_{R\uparrow\infty}
\sum_{i=1}^{d}E(\omega_{i,R},\mathbb{S}^2).
\end{equation}
Note that $\{\omega_{i,R}\}_{i=1}^{d}$ have uniformly boundedness of energies,
from Brezis-Coron \cite{BC1} one concludes that there exist
$N_i\in \Big[1,\frac {m}{\varepsilon_0}\Big]$ and $N_i$ bubbles
$\{\omega_{i,j}\}_{j=1}^{N_i}$ such that
\begin{equation}\label{j-sum-bubbles}
\lim_{R\uparrow\infty}E(\omega_{i,R},\mathbb{S}^2)
=\sum_{j=1}^{N_i}E(\omega_{i,j},\mathbb{S}^2).
\end{equation}
Now, putting all \eqref{i-sum-bubbles},\eqref{m-bubbles} and
\eqref{j-sum-bubbles} together, it follows that
$$
m=\sum_{i=1}^{d}\sum_{j=1}^{N_i}E(\omega_{i,j},\mathbb{S}^2).
$$
The proof of Theorem \ref{flow_identity} is complete.
\subsection*{Acknowledgements} This research is partially supported
by grant 11071012 from the NSFC, and by the program of visiting Chern Institute
of Mathematics. The author would like to thank Prof. Changyou Wang
for the valuable suggestions.
\begin{thebibliography}{00}
\bibitem{A} Adams, D. R.;
\emph{A note on Riesz potentials}. Duke Math J, 1975, {42} (4): 765-778.
\bibitem{BC} Brezis, H.; Coron, J.;
\emph{Multiple solutions of H-systems and Rellich's conjecture}. {Comm
Pure Appl Math}, 1984, {37 }(2): 149-187.
\bibitem{BC1} Brezis, H.; Coron, J.;
\emph{Convergence of solutions of H-systems or how to blow bubbles}.
{Arch Rational Mech Anal}, 1985, {9 }: 21-56.
\bibitem{BR} Bethuel, F.; Rey, O.;
\emph{Multiple solutions to the Plateau problem for nonconstant mean curvature}.
{Duke Math J}, 1994, {73}(3): 593-645.
\bibitem{ChL} Chang, K. C.; Liu, J. Q.;
\emph{An evolution of minimal surfaces with Plateau condition}. {Calc
Var}, 2004, {19}(2): 117-163.
\bibitem{CL} Chen, Y. M.; Levine, S.;
\emph{The existence of the heat flow for H-systems}.
{Disc Cont Dynam Syst}, 2002, {8}: 219-236.
\bibitem{CM} Caldiroli, P.; Musina, R.;
\emph{The Dirichlet problem for H-systems with small boundary data:
blow up phenomena and nonexistence results}. {Arch Rational Mech Anal},
2006, {181}: 1-42.
\bibitem{DG} Duzzar, F.; Grotowski, J. F.;
\emph{Existence and regularity for higher-dimensional H-systems}.
{ Duke Math J }, 2000, {101 }(3): 459-485.
\bibitem{DT} Ding, W. Y.; Tian, G.;
\emph{Energy identity for a class of approximate
harmonic maps from surfaces}. {Comm Anal Geom}, 1995, {3}: 543-554.
\bibitem{Hel} H\'elein, F.;
\emph{Harmonic Maps, Conservation Laws, and Moving Frames}.
Cambridge: Cambridge University Press, 2002.
\bibitem{Hil} Hildebrandt, S.;
\emph{On the Plateau problem for surfaces of constant mean curvature}.
{Comm Pure Appl Math}, 1970, {23}: 97-114.
\bibitem{HH} Hong, M. C.; Hsu, D. L.;
\emph{The heat flow for H-systems on higher
dimensional manifolds}. {Indiana Univ Math J}, 2010, {59}(3): 761-789.
\bibitem{HL} Han, Q.; Lin, F. H.;
\emph{Elliptic partial differential equations}.
Providence: Courant Press, 1997.
\bibitem{HTW} Huang, T.; Tan, Z.; Wang, C. Y.;
\emph{On the heat flow of equation of surfaces of constant
mean curvatures}. {Manuscripta Math}, 2011, {134}: 259-271.
\bibitem {LW1} Lin, F. H.; Wang, C. Y.;
\emph{Energy id entity of harmonic map flows from surfaces at finite
singular time}. {Calc. Var. Partial Differential Equations,} 1998,
{6}: 369-380.
\bibitem{MY} Mou, L.; Yang, P.;
\emph{Multiple solutions and regularity of H-systems.}
{Indiana Univ Math J}, 1996, {45}(4): 1193-1222.
\bibitem{QT} Qing, J.; Tian, G.;
\emph{Bubbling of the heat flows for harmonic maps from surfaces.}
Comm Pure Appl Math, 1997, {50}(4): 295-310.
\bibitem{Re} Rey, O.;
\emph{Heat flow for the equation of surfaces with prescribed mean curvature}.
{Math Ann}, 1991, {291}(1): 123-146.
\bibitem{St} Struwe, M.;
\emph{Nonuniqueness in the Plateau problem for surfaces of constant
mean curvature}. {Arch Rational Mech Anal}, 1986, {93}(2): 135-157.
\bibitem{Wente} Wente, H.;
\emph{An existence theorem for surfaces of constant mean curvature}.
{J Math Anal Appl} 1969, {26}: 318-344.
\bibitem{WZ1} Wang, C. Y., Zheng, S. Z.;
\emph{Energy identity of approximate biharmonic maps to Riemannian
manifolds and its application}. J Funct Anal, 2012, 263: 960-987.
\bibitem{WZ2} Wang, C. Y.; Zheng, S. Z.;
\emph{Energy identity for a class of approximate biharmonic maps
into sphere in dimension four}. Disc. Conti. Dyn. Syst. A, 2013, 33(2): 681-878.
\end{thebibliography}
\end{document}